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Full text of "Treatise On Analysis Vol-Ii"

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7 ALGEBRAIC PROPERTIES OF CONVOLUTION 275 Let St = Supp(jUj), and suppose that St is compact except possibly for one index j. Let/6 3CC(G) and let K be the compact support of/. It is enough to show that the set of points (xl9..., xn) e Gn which belong to the support n Y[ S,- of fa ® • * • ® jun (13.21.18) and are such that xvx2 ' * • xn e K, is relatively i=l compact in Gn. Now, the conditions xt e S£ for all /, and XiX2 •- xneK, imply that v- <= Q ~ * ...Q""lkrC~l...C~l X7 G °j- 1 ^1 ^^n ^j+ 1 J and this set is compact (12.10.5). Since by hypothesis St is compact whenever / ^67, our assertion is proved (3.20.16). The result of (14.6.2) or (14.6.3) shows that two measures can be convolv- able without either of them having a compact support. Later (14.10.7) we shall see examples of unbounded measures which are convolvable. 7. ALGEBRAIC PROPERTIES OF CONVOLUTION (14.7.1) Let A, /z, v be three measures on G, and suppose that the pairs (A, ju) and (A, v) are convolvable. Then so is the pair (A, ju + v) and we have (14.7.1.1) A * (fi + v) = A * /i + A * v. For by virtue of the relation \fi + v\ g |ju| + |v| we may restrict ourselves to the case where A, ju and v are positive, and in this case the result follows immediately from (13.16.1). Similarly, if (A, v) and (//, v) are convolvable, then so is (A + ju, v) and we have (14.7.1.2) (A + //)*v = A*v + /£*v. Also it is clear that if the pair (A, $ is convolvable, then so is (#A, fe/z) for all scalars a and b, and (14.7.1.3) (aX) * (bti=(ab)l * p. (14.7.2) Let A, ju, v be three measures ^0 on G. (i) J/if/ie sequence (A, /*, v) & convolvable, then so are the sequences (A, /*), | * |/4 v), Ox, v), (A, |M| * |v|), of/iJ we (14.7.2.1) A * ji * v = (A * /x) * v = A * (p * v).subset A of Nc is relatively compact in Nc if and only if it satisfies